1 IIT Delhi - CML 100:1 The shortfalls of classical mechanics Classical Physics 1) precise trajectories for particles simultaneous specification of position and momentum 2) any amount of energy can be exchanged implies the various modes of motion (translational, vibrational, rotational) can have continuous energies Light electromagnetic field frequency ν, wavelength λ, speed in vacuum c = 3 10 8 ms 1. λν = c wavenumber, ν = ν c = 1 λ Black Body An object capable of emitting/absorbing all frequencies of radiation uniformly Figure 1.1: Representation of a blackbody Figure 1.2: Change in the emission maxima from the blackbody as a function of temperature Rayleigh-Jeans Law Classical viewpoint electromagnetic field is a collection of oscillators of all frequencies Equipartition principle average energy of each oscillator = kt de = ρdλ ρ = 8πkT/λ 4 Ultraviolet catastrophe As wavelength decreases, the energy density increases to infinity
2 IIT Delhi - CML 100:1 The shortfalls of classical mechanics Planck s hypothesis All possible frequencies are not allowed energy of each oscillator is restricted to discrete values Quantization, E = nhν n = 0, 1, 2 where h is the Planck constant (= 6.626 10 34 J. s) de = ρdλ ρ = 8πhc λ 5 (e hc/λkt 1) Short wavelengths: hc/λkt 1 e hc/λkt faster than λ 5. So no UV catastrophe! The hypothesis fits well with the observed data. (Actually the value of h was determined by fitting) Why this works? Classical heat black body all oscillators with all frequencies excited the excitation of high frequencies leads to UV catastrophe Planck oscillators can be excited only if energy hν is available. Higher frequencies implies higher energies which is too large to supply by heating the walls of the blackbody Heat Capacities Dulong and Petit Law: Measured value for solids = 25 J K 1 mol 1 Mean energy of an atom as it oscillates about its mean position = kt for each direction (equipartiion principle). This gives a total of 3kT for the three directions. For a mole, U m = 3RT C V.m = ( U m T ) = 3R = 24.9 J K 1 mol 1 V At lower temperatures, molar heat capacities are lower than 3R. Einstein s explanation (1905): Each atom oscillates about its equilibrium position with frequency ν. Used Planck s hypothesis energy of oscillation equals nhν. This gives, U m = 3N Ahν e hν/kt 1 Differentiate w.r.t. T to get C V,m = 3Rf f = ( θ 2 E T )2 ( eθe/2t ) e θ E /T 1 Einstein temperature θ E = hν/k corresponds to the frequency of oscillation. At high temperatures (θ E T), expand f = ( θ E T )2 { which gives the high temperature result C V,m = 3R 1+ θ E/2T+ (1+θ E /T+ ) 1 }2 1 ignoring the higher terms 2 ) e θ E /T At low temperatures θ E T), f ( θ E T )2 ( eθe/2t zero than 1/T goes to infinity. = ( θ E T )2 e θ E/T Exponential decays faster to
3 IIT Delhi - CML 100:1 The shortfalls of classical mechanics Reason: At low temperatures fewer atoms have the energy to oscillate. Figure 1.3: Temperature dependence of heat capacities Further: All atoms do not oscillate with the same frequency. There is a distribution of frequencies ranging between 0 to ν D. Read Debye s formula for the heat capacities. Atomic and Molecular Spectra Figure 1.4: Molecular spectrum of SO 2. Figure 1.5: Energy levels in a molecule Radiation is absorbed/emitted in discrete steps and not continuously. This suggests the energy levels in an atom/molecule are quantized. The frequency of radiation absorbed/emitted is given by the Bohr frequency condition ΔE = hν Wave-particle duality Atomic and molecular spectra show that EM radiation of frequency ν can possess only the energies 0, hν, 2hν suggesting the radiation is composed to 0, 1, 2... number of particles called photons.
4 IIT Delhi - CML 100:1 The shortfalls of classical mechanics Photoelectric effect Ejection of electrons from a metal surface upon irradiation with light 1) No electrons are ejected unless the frequency of radiation is higher than a threshold value (work function). The intensity of radiation does not matter 2) The K.E. of ejected electron varies linearly with the frequency of the light used. It is independent of the intensity 3) All that the intensity does is to change the number of electrons ejected, i.e. the current Figure 1.7: The photoelectric effect Electron collides with a particle carrying sufficient energy required to eject it. Wave character of particles: KE = 1 2 mv e 2 = hν φ Davisson and Germer (1952) diffraction of electrons by a crystal. The basis of TEM and SEM. Diffraction requires constructive and destructive interference of waves. Hence, particles must possess wavelike character. de Broglie relationship (1924): λ = h/p. A very fundamental statement.
5 IIT Delhi - CML 100:1 The shortfalls of classical mechanics Hydrogen Spectra Figure 1.6: Emission spectrum of hydrogen atom on a log scale Figure 1.7: Atomic interpretation of the hydrogen spectrum Rydberg Formula: ν = 1 λ = 109680 ( 1 n 1 2 1 n 2 2 ) cm 1 (n 2 > n 1 ) Bohr Theory for H-atom: Electron revolving around the nucleus Coulomb s Law f = e 2 /4πε 0 r 2 ε 0 = 8.85419 10 12 C 2. N 1. m 2 Centrifugal force f = m e v 2 /r Bohr s assumption integral wavelengths fit the circular orbit for constructive interference 2πr = nλ which, on using the de Broglie equation gives
6 IIT Delhi - CML 100:1 The shortfalls of classical mechanics m e vr = nh 2π = nħ Equate the two forces and substitute the above expression for the first Bohr radius, a 0 a 0 = ε 0h 2 n 2 πm e e 2 = 4πε 0ħ 2 n 2 m e e 2 = 4π(8.85419 10 12 C 2. N 1. m 2 )(1.055 10 34 J. s) 2 (9.109 10 31 kg)(1.6022 10 19 C) 2 = 52.92 pm Total energy of the electron = KE + PE = 1 2 m ev 2 Using the above equations, E = e2 8πε 0 r And substituting for r, E n = m ee 4 8ε 0 2 h 2 1 n 2 e2 4πε 0 r And transitions between two states n 1 and n 2 will be given by or and the Rydberg constant R is given by m ee 4 ΔE = m ee 4 8ε 0 2 h 2 ( 1 n 1 2 1 n 2 2 ) = hν ν = m ee 4 8ε 0 2 ch 3 ( 1 n 1 2 1 n 2 2 ) 8ε 0 2 ch 3